A is the event that Red rolls $1, 2,$ or $3; B$ is the event that Red rolls $2, 4, or 6;$ and C is the event that the sum of the two rolls is 5.
(a) Find $p(A|B)$, $p(B|C)$, and $p(C|A)$
(b) Find $p(A|B ∩ C)$, $p(B|C ∩ A)$, $p(C|A ∩ B)$
(c) Are the three events pairwise independent? Mutually independent?
My attempt:
$p(A) = 1/2$, $p(B) = 1/2$, $p(C) = C(6,1)*4/6*C(6,1)*4/6*3/6$
$p(A ∩ B) = 1/2*1/3*1/2*1/3 = 1/36$
$p(B ∩ C) = 1/2*2/3*C(6,1)*4/6*C(6,1)*4/6*3/6$
$p(A|B) = (1/36)/(1/2) = 1/18$
$p(B|C) = p(B ∩ C)/p(C)$
I am lost after that point.
SOME HINTS:
See that $\{1,2,3\}\cap\{2,4,6\}=\{2\}$ so $P(A\cap B)=1/6$ (doesnt matter what happen on yellow dice i.e. $P(A\cap B)=6/36$ too).
The sum for two rolls 5 means that you can take $1+4$, $2+3$, $3+2$ or $4+1$ then $P(C)=4/36$.
Another example: $P(B\cap C)=2/36$ because the only way to sum 5 with $a+b=5$ where $a\in\{2,4,6\}$ and $b\in\{1,...,6\}$ is with $a=4$ and $b=1$ OR $a=2$ and $b=3$.
Event A is independent of B means that $A\cap B=\emptyset$ but this is not the case because $A=\{1,2,3\}$ and $B=\{2,4,6\}$.